Ok - an update...
I swapped the first unit for a new one today at the Apple Store. That did not fix the issue. Following some of the feedback in this thread I determined this is definitely some sort of BT interference issue related to my television.
My Apple TV sits on a shelf in a cabinet (technically in the other room) behind my TV. When I'm in front of the TV I have a range of <10 feet for my AirPods, the Siri remote, and any other bluetooth headphones. If I move into an adjacent room (so the TV is not directly between me and the Apple TV) I can get a range of 20-40 feet, with walls and doors in-between. I also tested the issue by moving my AppleTV to a shelf much lower in the cabinet, so that it would physically be lower than the height of my TV. This also solved the problem. It definitely seems to be some sort of issue with interference from the TV. I don't have the ability to discretely disable the BT or WiFi radios on the TV, so I can't test that specifically.
I did call Apple Support and logged a ticket with a senior tech. I gave him all the details and he acknowledged there were other similar reports. So it certainly seems to be something Apple is aware of and working on.
I'm curious if those of you still having this issue are in a similar situation. If you have any other devices with BT or WiFi right next to, or in-between, you and your AppleTV you might want to try moving things around to see if that will fix it. Even if it's not something that is going to work for you long term it would be interesting to know if this is the issue.